Problem: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{n^3 + 8n^2 + 12n}{n^3 + n^2 - 30n}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {n(n^2 + 8n + 12)} {n(n^2 + n - 30)} $ $ p = \dfrac{n}{n} \cdot \dfrac{n^2 + 8n + 12}{n^2 + n - 30} $ Simplify: $ p = \dfrac{n^2 + 8n + 12}{n^2 + n - 30}$ Since we are dividing by $n$ , we must remember that $n \neq 0$ Next factor the numerator and denominator. $ p = \dfrac{(n + 6)(n + 2)}{(n + 6)(n - 5)}$ Assuming $n \neq -6$ , we can cancel the $n + 6$ $ p = \dfrac{n + 2}{n - 5}$ Therefore: $ p = \dfrac{ n + 2 }{ n - 5 }$, $n \neq -6$, $n \neq 0$